Programme proposal
A. Title "QUANTUM INFORMATION THEORY AND QUANTUM COMPUTATION"
B. Programme Proposers
Prof. J.I. Cirac, Inst. fuer Theoretische Physik, Universitaet Innsbruck, Innsbruck, Austria
Prof. A. Ekert, Centre for Quantum Computing, Oxford University, Oxford, UK
Dr. M.B. Plenio, Imperial College of Science, Technology and Medicine, London, UK
C. Summary
The new concept of quantum computing which has been developed over the past few years promises immense computational power and new insights into quantum mechanics and information theory. Quantum algorithms have been discovered that allow the solution of practically important problems which cannot be solved on any classical computer. A new theory of quantum information is emerging. Possible experimental realizations of a quantum computer have been proposed, quantum gates and simple quantum algorithms have been demonstrated experimentally. The proposed network represents the combined effort of leading groups from 13 European countries to carry out theoretical and experimental studies in various applications of quantum computing as well as on fundamental concepts of the underlying quantum information theory. Research will focus on the following seven areas:
1. Theoretical and experimental study of quantum computing and quantum communication based on linear ion traps.
2.Theoretical and experimental study of quantum computing based on nuclear magnetic resonance schemes.
3.Theoretical and experimental study of the potential of quantum computing based on Josephson junctions.
4.Theory and applications of entanglement in quantum information theory.
5. Fundamental aspects of quantum algorithms.
6. Decoherence on a quantum computer and quantum communication.
7. Investigation of the potential of hybrid quantum computing-classical information technology and the development of its implementation.
8. European context: R & D collaborations which already exist in the field
9. European added value
10. Envisaged operational programme actions.
This network comprises groups that are active in the above areas and have achieved important theoretical and experimental results in all of them. The programme will allow us to enhance existing successful collaborations, as well as to establish contacts with newly formed groups and with some not yet involved in existing networks.
In addition some of our objectives are: a) faster dissemination of scientific results;
b) sharing of experimental skills and equipment;
c) closer interaction between theoretical and experimental groups;
d) coordinated education of young scientists.
We will implement a scheme of bilateral visits, fellowships, conferences and joint publications. In particular in the light of the substantial support that this field obtained recently in the USA the present network would help to strengthen the leadership of Europe in this newly emerging and highly active field.
D. Status of the research context and envisaged achievements of the programme
In recent years the new fundamental concept of quantum computation and quantum information theory has been developed (1, 2, 3, 4). Instead of using classical bits that can represent either the values 0 or 1, the basic unit of a quantum computer is a quantum mechanical two-level system (qubit) that can exist in coherent superpositions of the logical values 0 and 1. A set of n qubits can then be in a superposition of up to 2n different states, each representing a binary number. Computations are implemented by unitary transformations which act on all states of a superposition simultaneously. Quantum gates form the basic units from which these unitary transformations are built up.
The use of the quantum mechanical superpositions and entanglement results in a high degree of parallelism which can increase the speed of computation exponentially. A number of problems which cannot be tackled on a classical computer can be solved efficiently on a quantum computer. In 1994 a quantum algorithm was discovered which allows the solution of a practically important problem, namely factorization, with such an exponential increase of speed [5]. Subsequently, possible experimental realizations of a quantum computer have been proposed, e.g. in cavity quantum quantum electrodynamics (QED) [6], linear ion traps [7] and nuclear magnetic resonance (NMR) schemes [8; 9]. Presently we are at a stage where quantum gates have been demonstrated in ion traps [10, 11], entangled atom states have been prepared using cavity quantum electrodynamics [12] and simple quantum algorithms have been demonstrated by NMR schemes [22, 23, 25, 26, 27]. Quantum computation is closely related to quantum cryptography [13, 14] and quantum communication. Basic experiments demonstrating the practical realizability of these ideas have been carried out to demonstrate quantum cryptography [15, 16], the violation of Bell inequalities over large distances [17] and quantum teleportation [18] as an important ingredient of quantum communication.
The participating research teams in this proposed network have made major contributions to the experimental and theoretical study of quantum computing, communication and information theory. We plan to extend this successful work in a number of directions which cover the main directions of growth in this active field. It is expected in several of these areas that major advances can be achieved within the time frame of this programme.
Although the proposal is split into seven different parts, we would like to stress that these parts are strongly related. This will ensure that the network will foster strong interactions not only between scientist in the same field but also between different disciplines.
1) Theoretical and experimental study of quantum computing and quantum communication based on linear ion traps.
Status: The linear ion trap is one of the most promising systems for quantum computation on five to ten qubits, and possibly more [19, 20]. The quantum state preparation (laser cooling and optical pumping) in this system is a well-established technique, as is the state measurement by electron shelving and fluorescence. The method to perform quantum processing in linear ion traps was discovered by members of this network [7], and several experimental groups within this network are currently working to implement it. Singly-charged ions of an atom such as calcium or beryllium are trapped and laser-cooled to micro Kelvin temperatures, where they form a string lying along the axis of a linear rf Paul trap. The internal state of any one ion can be exchanged with the quantum state of motion of the whole string, by illuminating the ion with a laser pulse at a frequency tuned below the ion’s internal resonance by the vibrational frequency of one of the normal modes of oscillation of string. This couples single phonons into and out of the vibrational mode. The motional state can then be coupled to the internal state of another ion by directing the laser onto the second ion and applying a similar laser pulse. In this way general transformations of the quantum state of all the ions can be generated.
The ion trap has several features to recommend it. It can achieve processing on more than five quantum bits without the need for any new technological breakthroughs, such as microfabrication techniques or new cooling methods. The state of any ion can be measured and re-prepared many times without problem, which is an important feature for implementing quantum error correction protocols. The trapped ions can be strongly coupled to an electromagnetic field mode in a cavity, which permits the powerful combination of quantum processing and long-distance quantum communication [21]. Once mastered, ion trap experiments can be miniaturised by microfabrication techniques. The upper limit to the number of qubits which ion traps might manipulate is unknown, but these systems can almost certainly realise a quantum processor larger than any one that could be thoroughly simulated by classical computing.
Aims: The initial aim will be to realise one or more quantum processors capable of manipulating three or more quantum bits. This will permit the demonstration of fundamental concepts such as quantum algorithms, quantum error correction and of other applications that will be developed in the framework of this network (see parts 4 and 5). The next stage is to push the experiments towards ten quantum bits. In parallel we plan to investigate the coupling of an ion trap processor to a high-quality optical cavity. This would then allow to map the electronic states which represent the qubit into the cavity lightfield. These photons could then be sent down an optical fibre to a different cavity - ion trap system [21]. This method would be the first building block of a quantum communication network with linear ion trap nodes. The theoretical work in this direction will be performed by groups in this network and are detailed under point 4 of this proposal.
Besides realising more and more advanced quantum processing, the aim of this research is to understand decoherence in such a complicated quantum system, and discover what are the ultimate limitations of the method. These experimental investigations will go hand in hand with theoretical work as detailed under point 6 of this proposal.
2) Theoretical and experimental study of quantum computing based on nuclear magnetic resonance schemes.
Status: During the last few years progress in the implementation of small nuclear magnetic resonance (NMR) quantum computers has been extremely rapid [8, 9]. Two-qubit quantum computers have been built, based on both homonuclear and heteronuclear spin-systems, and have been used to demonstrate a range of quantum computational processes, including algorithms to solve Deutsch’s problem [22, 23] and quantum searches based on Grover’s algorithm [24, 25]. Three spin systems have also been investigated, and some basic three qubit processes have been demonstrated [26]. Researchers from Oxford who are part of this network have made pioneering work in this area. It will be important to ensure that their knowledge is passed on to other NMR groups within the network to ensure that Europe remains competitive in NMR quantum computing.
Aims: In the short term research will concentrate on the development of NMR quantum computers with three to six qubits. Such computers could be used to investigate algorithms such as quantum error correction, and even the implementation of Shor’s quantum factoring algorithm applied to small numbers. While three qubit computers appear to be relatively straightforward, larger systems are likely to prove experimentally challenging. The reasons for this are twofold. First, the signal obtainable from an NMR quantum computer decreases exponentially with the number of qubits in the computer. We plan to investigate approaches that tackle this problem. Second, with increasing numbers of qubits it will become difficult to selectively address individual nuclei, but it should be possible to reduce these difficulties by applying selective excitation techniques developed in other areas of NMR.
Another problem is the inevitable decoherence in NMR experiments. Although the decoherence times can be of the order of seconds this will impose limitations to the number of quantum gates that can be executed on an NMR quantum computer. We plan to investigate methods to overcome this limitation, by using larger spin--spin interactions, such as dipolar couplings in the solid state, to reduce the gate time, or by applying the techniques of quantum error correction to NMR (see also part 6). A quite different approach to solving these problems is the investigation of quantum computing based on single trapped molecules. While the experimental difficulties involved are formidable, this approach would permit fruitful interactions between NMR designs and more conventional quantum computers based on trapped ions.
Current research has concentrated on the implementation of quantum computers which emulate ion trap models. In the longer term research is likely to move towards new designs of NMR quantum computers which take the strengths and weaknesses of NMR into account. For example, the ensemble average implicit in NMR measurements allows readout techniques quite unlike those used in other designs. In particular the possibility of directly probing superpositions should greatly simplify some problems. We plan to investigate the potential of these new features in NMR schemes. This will be done in close collaboration between experts in NMR computing and researchers interested in `traditional’’ quantum algorithms.
Apart from NMR schemes it will also be interesting to assess the possibilities offered by related fields such as ESR ({em electron spin resonance}), NQR (nuclear quadrupole resonance) and combination approaches such as ENDOR (electron nuclear double resonance) [28]. Like NMR these are highly developed techniques which are widely applied in physics, chemistry, and biology.
The aims outlined in this part of the proposal allow for interdisciplinary work between network participants including NMR experts, ion trap experts, theoretical physicists interested in decoherence and quantum error correction, and computer scientists working on the development of interesting new algorithms and concepts.
3) Theoretical and experimental study of the potential of quantum computing based on Josephson junctions.
Status: Recently schemes for using Josephson junctions to perform quantum logical operations have been proposed [29, 30]. These schemes are based on the dynamics of Cooper pairs in nanoscale tunnel junctions of a superconducting metal. For industrial purposes it is of great importance to find a quantum computer model based on reproducible solid-state components.
In a superconductor, electric current is carried by Cooper pairs without resistance. When a small superconducting island is separated by tunnel junctions from other electrodes, the number of Cooper pairs determines the energy of the state of the system in form of a classical charging energy. If the island and the junctions are made very small, the electrostatic energy of the system can easily exceed any other available energy, the thermal energy in particular. Then, by electrostatic modulation of the island potentials using the adjacent gate electrodes, one can make electron pairs (= Cooper pairs) to tunnel into or from the island. Single electron tunnelling is prohibited due to the energy gap of the superconductor.
If one fabricated a row of tunnel junctions and islands with gates in series, Cooper pairs could be moved around within the array by controlling the gate voltages [29]. Because the population n of the islands is typically 0 or 1, with properly chosen gate voltages a pair of islands can be considered as a qubit where the 0 corresponds to the neutral state of the islands and 1 corresponds to a state where a Cooper pair has been moved from one island to the second one. These qubits can be moved such that the islands forming the qubits sit next to each other, or they can be moved apart from each other. In the first case tunnelling between these islands is probable, in the latter case, however, the tunnelling does not occur.
In a simple two-bit operation, the control not gate (CNOT), one has a control bit and a target bit. The gate operation simply inverts the state of the target bit if the control bit has the value 1, and does nothing if the control bit has the value 0. In the case of qubits, if the control bit is in a superposition state, then the target bit ends up in a superposition state as well, and now both bits become entangled. In a Cooper pair computer, the difference between the control and target bits lies in the coupling between two more or less similar arrays of small Josephson junctions. Two arrays can couple capacitively if they are fabricated sufficiently near to each other. The state of the control bit can be rigid enough if one applies gate voltages, but the capacitive coupling between the arrays can be used to change the energy relations between the two states of the target qubits sufficiently, to allow logic operations.
Aims: The group at the University of Jyväskylä (Finland) has started an experiment which aims at realising quantum gates with Josephson junctions. The general aim of this project is to study the possibility of performing quantum gate operations between quantum bits coded into Josephson junctions. The ultimate aim is to demonstrate quantum gate operations following the scheme proposed in [29]. However, it is also beneficial to study the occurrence and survival of quantum coherences in these very popular solid state systems. As nanostructures are becoming increasingly popular and common, it is important to uncover the basic quantum mechanical properties of the Josephson junction systems.
In the beginning the work is carried out to demonstrate experimentally the elements of Cooper pair qubits, to simulate theoretically the manipulation of Cooper pairs, and to develop the niobium based junction technology. The research team aims to demonstrate by theoretical means that the qubits of the type suggested can be maintained and manipulated, and that the logic operations are feasible in the realistic experimental conditions. Also, they intend to demonstrate experimentally that elementary single Cooper pair systems like a superconducting single electron transistor, a single Cooper pair box and a two island Cooper pair pump, can be operated. The first two are being investigated both in Jyväskylä and elsewhere already now [31]. If all this can be achieved successfully, then the next aim is to demonstrate experimentally the CNOT gate using Cooper pairs as qubits.
The experimental work at Jyväskylä connects well with the theoretical activities within the proposed network. In the solid state systems noise has a clear role, which needs to be properly studied, using e.g. the approaches discussed in part 6. The application of the studies of entanglement described in part 4 to solid-state systems becomes an important challenge if the demonstration of a gate operation with Josephson junctions is successful. The network offers an excellent forum for developing exchange of information and development of collaboration between the solid-state physics experimentalists and the quantum information theoreticians.
4) Theory and applications of entanglement in quantum information theory.
Status: The recent development of quantum information theory has shown that entangled states allow for new forms of information processing. Namely, entangled states are the key-ingredient for certain forms of quantum cryptography [14, 15] and for quantum teleportation [18, 32], which has been demonstrated experimentally by participants of this network. Entanglement is also responsible for the power of quantum computing which, under ideal conditions, can accomplish certain tasks exponentially faster than any classical computer [5].
Therefore it is of greatest interest to examine quantum entanglement and its applications in physics and quantum information theory in detail. A deeper understanding of the role of quantum entanglement in quantum information theory will allow us to improve existing applications and develop new methods of quantum information manipulation. This progress will eventually lead to technical applications of the new methods developed in this field.
Aims: In an application that requires quantum entanglement it is of major importance that we have pure maximally entangled states at our disposal. For example, in teleportation we need to establish a pure maximally entangled state between sender and receiver of a quantum state. If one attempts to establish such a maximally entangled state, then noise during transmission will generally lead to partially entangled mixed states, e.g. Werner states [33]. The task is then to obtain from a set of these partially entangled states fewer maximally entangled states using only local operations (only acting on one side of the entangled pair) and classical communication. In fact such techniques, called quantum state purification methods, exist [34, 35, 36, 37, 38]. However, currently proposed methods of this kind are extremely difficult to implement experimentally and are certainly not optimized. Therefore we plan to investigate quantum state purification methods in detail. The aim is to provide methods that are more feasible experimentally. We also plan to optimize the efficiency with which precious resources such as entangled states are consumed. In many experiments, such as for example in the transmission of photons through an optical fibre a particular kind of noise will be dominant. Therefore we plan to investigatewhether it is possible to develop specially adapted purification schemes which are then more efficient than general schemes. We also plan to investigate methods in which we ’pump’ entanglement from a supply of weakly entangled states in to a fixed pair of particles. Such a scheme would be of particular interest as it is naturally adapted to experimental situations given in quantum communication experiments. These investigations support in particular the development and practical implementation of quantum communication networks based on combined ion trap - optical cavity nodes which are connected by optical fibres as they are investigated experimentally in part 1 of this proposal.
The efficiency of purification schemes is closely related to the question of the amount of entanglement in a given quantum state. The more entanglement a state contains, the more maximally entangled states one can obtain using quantum state purification. Measures for the amount of entanglement have been proposed recently [39, 40, 41, 42, 43, 44], and we plan to investigate their mathematical structure as well as their physical implications in particular with respect to the efficiency of quantum state purification procedures and the problem of quantum channel capacities. Apart from quantitative approaches to mixed state entanglement there is still a need for qualitative investigations. A general algorithm allowing to determine whether a given quantum state is entangled has been found only for two-qubit and qubit-trit systems [45, 46, 47, 48].Therefore we plan to continue the search of operational characterization of the set of entangled mixed states for pairs of many-level systems. The qualitative approach to the purification of mixed state entanglement is also required. The known protocols of purification are not only inefficient but also they are not universal. For two-qubits systems there is a protocol capable to purify any mixed entangled state [38]. In the case of higher dimension, we still do not know which mixed states can be purified to pure maximally entangled states. Moreover, quite recently a kind of entanglement has been discovered [49] which cannot be converted into pure entanglement at all. In this context the fundamental problem is to determine the role of the non-purifiable entanglement for quantum communication theory. Another basic task is to determine which entangled states can be purified (this was initiated in [50]). These investigations will have important practical implications and they will also help us in gaining a deeper understanding of entanglement and entanglement manipulations.
While the aims described above deal mainly with two-particle entanglement it is interesting (and important for the following) to consider the case of multi-particle entanglement [51, 52, 53, 54]. We therefore plan to investigate both fundamental properties as well as potential applications of multi-particle entanglement. Again it will be extremely important to be able to generate pure maximally entangled multi-particle states [55]. Therefore we plan to investigate and optimize multi-particle purification procedures.
In addition to the thorough investigation of quantum state purification methods we also plan to investigate potential new applications of entanglement in systems consisting of few two-level systems. Such applications require only moderate experimental resources and are therefore more easily implementable than complicated algorithms such as that for the factorization of large numbers. A particular example is the use of entanglement to improve optical frequency standards, i.e. atomic clocks [56, 57]. Initial steps in this direction have been done by members of this network [57]. In the framework of this network we plan to investigate whether quantum error correction methods can lead to more precise atomic clocks. Another application of entanglement lies in distributed quantum computation [58]. It has recently been shown by members of this network that certain computations can be performed more economically when spatially distributed parties use maximally entangled states in the course of the computation [59] even in the presence of noise. In close collaboration with the computer scientists in this network we plan to investigate for which classes of computations this result applies and which advantages can be achieved (see also part 5). These are just two examples of possible small scale applications of entanglement and we plan to investigate and develop more of these methods.
All these investigations will improve our understanding of entanglement and entanglement transformations and will assist in the formulation of a ’thermodynamics of entanglement’. Just as thermodynamics provides us with a intuitive understanding of complicated processes of statistical mechanics such a formulation would help us to get an intuitive feeling for all processes that involve the manipulation of quantum entanglement. Such an understanding would greatly assist the development of the theory of entanglement in quantum information theory.
5) Fundamental aspects of quantum algorithms.
Status: The development of small scale quantum algorithms -- which might be implemented experimentally in the shorter-term - is clearly of great importance and interest. However the study of quantum algorithms in a wider context and on a larger scale, is also of great theoretical interest, particularly for fundamental physics. It addresses an essential new difference between quantum and classical physics which has been revealed by the novel information-theoretic perspective of our subject, i.e. the intrinsic difference in the complexity of time evolution in quantum physics as compared to classical physics [60]. One may expect that many of the major {em long} term benefits of the subject will rest on developments of this type, which are not necessarily motivated by immediate experimental possibilities. In this area we plan to investigate fundamental aspects of quantum algorithms with the aim of a better understanding of quantum computation. It will also be of importance to investigate the effect of decoherence on the implementation of these quantum algorithms [[61, 62] (see also part 6). These problems allow for collaborations between computer scientist, theoretical physicists interested in the problem of decoherence and experimentalists in the field of ion traps, NMR quantum computing and Josephson junctions.
Aims: All of the known quantum algorithms [1, 5, 24, 63, 64, 65, 66] rest on a few basic principles, most notably the use of the Fourier transform on an Abelian group to extract periodicities [67]. To find new algorithms we propose initially to investigate the possible generalisation of these principles to other Abelian groups and further afield to non-Abelian groups. A natural candidate for study here is the so-called Graph Isomorphism problem - much studied in classical computation -- for which no efficient algorithm is known. This problem has a natural underlying non-Abelian symmetry, for which an efficient quantum Fourier transform has already been developed [68]. However the workings of the Fourier transform in a non-Abelian setting appear to be quite different from the Abelian case and despite its tantalising formal similarity to known algorithms the Graph Isomorphism problem has so far resisted solution.
In addition to extending the remit of known techniques, we will seek to identify new principles for quantum algorithms. Recent work [69, 70] by members of the network suggests strongly that computational speedup in quantum algorithms cannot be obtained without addressing in a significant way, the particular mathematical structure of the problem at hand i.e. it does not suffice to treat the input as a ``black box’’ or oracle and merely compute standard properties in (possibly exponentially large) superpositions. Thus for example, the famous algorithms of Simon [65] and Shor [5] utilise the mathematical structure of periodicities, which turns out to fit remarkably well into the formalism of quantum computation (as the Fourier transform is conveniently, always a unitary matrix. We will aim to identify other mathematical structures which might naturally complement the formalism of quantum physics to provide a basis for new quantum algorithms.
In addition we plan to investigate the problem of computational complexity. It has been shown that using experimentally verified quantum mechanical effects (as resulting from the so-called Einstein-Podolsky-Rosen paradox) one can distributedly compute exchanging less classical bits in communication than is possible purely classically [71]. This result is absolute and does not leave open the possibility of competing purely classical methods. These results have been improved in cooperation with other researchers to obtain an exponential gap between classical and quantum communication complexity for certain problems [73]. It will be important to continue these investigations to identify practically important problems for which we can obtain advantages over classical computations. It is important to establish whether these improvements persist in the presence of noise due to an environment [59]. This connection to decoherence (see also part 6) opens up the opportunity for an interdisciplinary collaboration between computer scientists, theoretical and experimental physicists.
We will also aim to approach the question of new principles of computation from a physical point of view [74]. The computational speedup in all the known quantum algorithms may be traced to various properties of quantum entanglement. Thus we can begin by looking at other non-classical aspects of quantum theory and ask if they can be used to provide novel computational possibilities.The development of any new principles for quantum algorithms is expected to have a very great long term impact on the subject.
6) Noise in quantum computing and quantum communication.
Status: The new effects and possibilities of quantum information theory depend crucially on the generation and preservation of quantum mechanical entanglement between many quantum bits. Decoherence, however, tends to destroy entanglement and in fact, highly entangled states are particularly sensitive to decoherence. It has been shown that noise in quantum computing is a serious problem [61, 62]. Subsequently, however, methods of quantum error correction have been developed [75, 76, 77, 78, 79, 80, 81, 82] and it has been shown that in principle they can solve the noise problem in quantum computing and quantum communication. Nevertheless many questions remain open and a more detailed analysis of the noise problem would be of great value both for the practical implementation of quantum computing and quantum communication as well as for our understanding of quantum noise in general. It is therefore of paramount importance to investigate the impact of decoherence on our ability to manipulate quantum information and to develop methods that allow to correct for errors due to decoherence.
Aims: To study the impact of decoherence especially in a quantum computer we plan to develop methods to simulate special properties of a quantum computer efficiently. The approach will be interdisciplinary and we plan to adapt techniques for example from solid-state, nuclear physics and quantum chemistry.
For several decades, a variety of analytical and numerical techniques have been developed for predicting the behaviour of quantum many-particle systems in numerous physical contexts, such as nuclear physics, condensed matter physics, etc. (see e.g. [83]. It is therefore worthwhile exploring the applicability of this body of knowledge to the simulation of quantum circuits. In particular, it is interesting to investigate the use of Monte Carlo techniques developed in the context of the numerical study of quantum many-body systems. Quantum computers can be seen as peculiar quantum many-body systems that evolve according to a non-local time-dependent interaction so as to carry out a {em computation}. The component quantum bits interact via a sequence of quantum gates, performing each a prescribed unitary transformation. Two-bit (or, more generally, n-bit) gates therefore effect non-local interactions between qubits, and the quantum algorithm (characterized by a network of quantum gates) corresponds to a specific sequence of unitary transformations, i.e., a time-dependent interaction.
In previous work [84], members of this network have investigated a many-body approach to quantum computation based on the Hubbard-Stratonovich transformation. This method can reduce the many-body dynamics of the N -bit quantum computer in a 2N -dimensional Hilbert space to the time evolution of N noninteracting quantum bits in auxiliary fields. Solving the dynamics of the quantumcomputer then amounts to evaluating a high-dimensional - but polynomial in N - integral over auxiliary fields. The latter can be approximated by use of a stochastic method.
This Monte Carlo technique has been applied to the quantum circuit for a 2-bit Quantum Fourier Transform [84]. However, many questions such as the scaling behaviour of these simulation techniques for large N remain open. In particular, unless the so-called ``sign-problem’’ (common to many Monte Carlo simulations) can be overcome, the standard Metropolis method seems to be inefficient in the context of quantum computation. It is therefore necessary to investigate several alternative approaches.
As decoherence is unavoidable in a quantum computer we plan to continue the investigation of quantum error correction codes [75, 76, 77, 78, 79] to which members of this network have made significant contributions. In these methods one quantum state is being entangled with some auxiliary quantum states. If one of these quantum states suffers an error then it is possible to reconstruct the original quantum state from the information in the remaining undisturbed quantum qubits.
In the framework of this network we plan to investigate quantum error correction methods in detail. While codes can be found that are able to deal with arbitrary kinds of errors in a particular experimental situation certain kinds of errors will usually be dominant. Therefore it would be desirable to develop methods of quantum error correction that are especially adapted to the correction of these predominant errors. Codes that are adapted to special error models will be much more economical with resources such as the number of required qubits. While traditional quantum error correction methods are adapted to qubits it is important to generalize these methods to be able to correct quantum systems with continuous degrees of freedom. This would allow the correction of errors for example in optical wavepackets. These methods which have been pioneered by a member of this network [85, 86], offer the potential to realize long distance quantum communication systems.
Error correction itself constitutes a quantum computation and is therefore susceptible to errors. However, very recently ’fault-tolerant’ quantum error correction methods have been developed [79, 80, 81]. These methods can tolerate errors in the error correction and it has been shown that once a certain error rate per gate operation can be achieved, arbitrarily long quantum computations can be performed. However, currently the required error rates are extremely low. We plan to investigate how these methods can be improved to allow for larger errors. Fault-tolerant methods may be adapted to special experimental scenarios of practical interest which would increase their efficiency substantially. It is both, of theoretical as well as of practical interest to determine what the maximally tolerable error rates in these fault tolerant methods are. We plan to explore the ultimate limits of these techniques will be explored. We also plan to underpin these investigations by an abstract investigation of quantum error correction methods using an information-theoretic approach to entanglement [87]. Such an approach will allow us to gain limits to the performance of quantum error correction methods. It will also give us new insights into the structure of these quantum error correction methods. This knowledge will help us in advancing applications of quantum error correction codes.
This part of the proposal will use techniques and experience gained in the other areas of the proposal and from a large number of participants of the network (see in particular part 4). It will of particular importance for the development of the practical implementation of quantum computing and quantum communication (see parts 1-3).
7) Investigation of the potential of hybrid quantum computing-classical information technology and the development of new implementations of quantum computing
Status: One of the most successful approaches from classical information technology (IT), i.e. silicon- based computers is a method called layering. Conventionally, it starts at the bottom with the device (or physical) level, then up to the gate and circuit levels, arriving at the highest levels of abstraction as the program, symbol and knowledge levels. The power of levels, which are basically modelling levels, comes from the following features: a) any system can be described at each of these levels independently; b) any level can be reduced to the level below (mapping); c) each level has not only a descriptive power but also a predictive power.
Aims: We aim to develop a methodological and technological framework for hybrid computing that combines classical silicon-based IT, with quantum computing (QC). Such an approach would offer the potential combination of advantages of both classical and quantum mechanical computer to improve computational devices. We will focus on those aspects that are essential to hybridization. Our goal is to establish a series of tools and methods that allow for constructing by routine and accountable means a physical system that realizes certain specifications. These investigations that will be conducted in close interaction with the other parts of this proposal will also assist in the development of new implementations of quantum computation.
We will investigate whether it is possible to transfer the methodological paradigm of layering to the technology of QC, and use it as a means to connect both with the aim of future hybrid applications. Therefore on one hand we want to study the corresponding levels that apply to QC, their notation and relations. On the other hand, we also aim to identify the levels of description that are common to both IT and QC. It is at these levels that a hybrid methodology can originate. It will be studied how a problem can be formulated at the common levels, decomposed and mapped along two paths (the IT and the Quantum one) down to a physical realization (silicon level and quantum level) of a truly coupled hybrid computer. While the physical realization of conventional IT is well defined, currently there are many physical systems that can be the basis of QC. One of our goals is to study the different possibilities and to establish the advantages and disadvantages of each of them.
We plan to proceed along the following lines and aims:
- Descriptive layers for QC: an account of different levels of abstraction, their delimitation, and their commonly used or newly developed notational conventions;
- Constraints and mappings: a systematic account of the constraints between the different layers, and the way these are reflected in prescriptive rules;
- Hbridization: a description of the methodological approach to designing a hybrid QC-IT application, by problem formulation at higher levels and abstract and systematic mapping to complementary and coupled realizations in both underlying technologies;
- Road map: an identification of the different gaps, risks and options in the hybrid approach;
- Limited experimental work towards a practical realization of a hybrid IT-QC computer;
- Writing and testing of software both for a reference application of database retrieval and for compilation, simulation and closed loop computing;
- A description of a possible realization of a hybrid IT-QC computer should be written. We would also like to submit a patent application if possible.
We plan to produce written documents to each of these points which will consist of a general discussion, a series of examples and, wherever possible, a report on the state of the art for its technical realization. The forum of the other areas of this ESF programme will be used for discussion as well as for theoretical and, wherever possible, experimental validation.
8) European context: R & D collaborations which already exist in the field
Some of the teams involved have already established links within EU TMR Networks devoted to "The physics of quantum information" (Contract ERBFMRXCT960087) and "Microlasers and Cavity Quantum Electrodynamics" (Contract ERBFMRXCT960066). Further collaborations exist under the ESPRIT Basic Research of excellence programme 7360: Physics and Technology of Mesoscopic Physics (PHANTOMS) as well as an EPSRC network on quantum optics within the UK and a Schwerpunktprogramm of the DFG in Germany. However, the proposed ESF programme covers more countries (13) and groups (around 30) than any of the networks mentioned above. Moreover, it has a strategic focus on cross-fertilising between experimental and theoretical skills from different disciplines of physics, chemistry and computer science, an aspect that is much less pronounced in the above mentioned networks.
In addition, the development of quantum information theory has been rapid and has led to new promising ideas for the implementatios of a quantum computer (see part 2 and 3) as well as new insights into possible applications (see part 4) and the structure of quantum information theory (see part 5). These developments go beyond the scope of existing networks. The proposed programme would provide valuable support to investigate these new developments together with the new groups that have entered the field. The proposed programme also includes two industrial partners which will facilitate the development and potential exploitation of new technical applications arising from quantum information theory.
At present the ESF has no activity that is centered on the topics proposed here. The EU has already expressed interest in quantum computing by establishing the Pathfinder project within Long Term Research of ESPRIT, which is not a research project but rather aims at charting the field in preparation of the Fifth Framework Programme. The major research groups within Pathfinder are also involved in the present proposal.
9) European added value
The study of the new concept of quantum computation and the underlying theory of quantum information is, as a new branch of physics, important both for our fundamental understanding of physics and information theory, as well as for the applications that will emerge. This programme involves the European groups that are playing a leading role in establishing both the theoretical and the experimental foundations of this field. One of our goals is to strengthen the collaboration between them, but also to incorporate newly formed groups and those from outside the EU and existing collaborations. In addition some of our objectives are: a) disseminate scientific results in the fastest possible way; b) share experimental skills and equipment; c) enhance the interaction between theoretical and experimental groups; d) enhance the education of young researchers and promote interaction between them; e) develop links across a broad interdisciplinary spectrum of researchers from physics, computer science, chemistry and industry. Therefore the programme represents a major step towards a unified and coordinated scientific effort in this field all over Europe. This will establish a broader and more solid basis for research in quantum computing and quantum information theory. In particular in view of the substantial support that this field currently experiences in the USA we believe that only through this support Europe will be able to maintain its competitive edge and its world-leading position in this field.
10) Envisaged operational programme actions.
1.Individual short scientific visits
In order to exchange knowledge, research results and to foster joint publications we plan short visits (of about one week) of young and senior researchers to other teams of this network (around 2 per team per year).
2. Short fellowships for young researchers
To help younger researchers to acquire knowledge and experience we plan to award short fellowships of about three weeks (10 per year) which will be given to pre- or post-doctoral researchers within this network. This will give them the opportunity to work with leading research teams and will enhance future fruitful collaborations.
3.Conferences
One conference (around 65 participants) will be organized per year, bringing together the different groups. At these meetings new results will be presented and sessions of the special workgroups will be organized.
4. Publication of Proceedings
In addition to joint publications in refereed journals it is planned that this network produces a book on quantum information theory. This will take place towards the end of this network, summarizing the activities of the network in such a form that it can be used as a textbook for new students in the field.
